Introduction to Finite Element Analysis and Design, 2nd Edition

Chapter 6Finite Elements for Two‐Dimensional Solid Mechanics

6.1 INTRODUCTION

All real‐life structures are three–dimensional. It is engineers who make the approximation as a one‐dimensional (e.g., beam) or a two‐dimensional structure (e.g., plate or plane solid). In chapter 5 we explained in detail the conditions under which such approximation could be made. When the stresses on a plane normal to one of the axes are approximately zero, then we say that the solid is in the state of plane stress. Similarly, when the corresponding strains are zero, the solid is in the state of plane strain. A two‐dimensional solid is also called a plane solid. Some examples of plane solids are: (1) a thin plate subjected to in‐plane forces; or (2) a very thick solid with a constant cross‐section in the thickness direction. In this chapter, we will discuss when an engineering problem can be assumed to be two‐dimensional and how to solve such a problem using two‐dimensional finite elements. We will introduce three different types of two‐dimensional problems and corresponding two‐dimensional elements. Every element has its own characteristics. In order to use the finite element method appropriately, a thorough understanding of the capabilities and limitations of each element is required.

In general, two‐dimensional elasticity problem can be expressed by a system of coupled second‐order partial differential equations. Based on the constraints imposed in the thickness direction, a two‐dimensional problem ...